216 FIGURE OF THE EARTH. [11 



Substitute for V; then 



Now let K Q , e , refer to the surface, so that 



r=/c 



substitute for r, neglecting e when multiplied by the small quantities E or 

 w 2 ; then 



this must be true for all values of X; we get the two equations 



the latter equation gives e in terms of E and w. 



Now gravity at the Earth's surface is the resultant of the attraction of 

 the body of the earth and the centrifugal force ; that is to say, at any 

 latitude, writing g for gravity, 



M 3E /. 1 



, 



M 2 , j M 3 2 1 / 1\ 

 = J 7,-o w ' < o+S2- 2 e -- T +w/cJ X --). 



K O ( K K J \ OJ 



Eliminate E by means of the relation found above ; we find 

 M 2 /5 M \A. IN 



Ml 1 \ 3 /5 2 



Let 6r denote gravity at the equator, where X = ; then 



Ml 1 \ 3 



Also the centrifugal force at the equator is o> 2 /c (!+-) ; let < be the 



V 8 / 

 ratio this bears to 6r ; then 



IT/ 1 \ 3 2 M 



-Y(l + o e o)-oX 

 K \ O / A 



to the range of approximation we have used. 



